How can you make "cross-referenced" "numbered items" in Word update when referencing paragraph numbers -- manually is perfectly fine?

Cross-references, like many fields in Word, are not auto-updating. Ref fields should update upon printing or print preview. See this Microsoft page on which fields get updated, when.

Try Cmd+A followed by Fn+F9.

Here is the Microsoft page on Keyboard Shortcuts.

Here is my chart of keyboard shortcuts dealing with fields.

You are not using the correct shortcuts. Cmd+Fn-F9 should create a new field, not update a current one!

Here is a macro that should update only cross-reference fields, wherever located in the document.

Sub RefFieldUpdateAllStory()
'   Written by Charles Kyle Kenyon 15 November 2001
'   repaired by Jezebel
'   All Story Field Updater - Ref fields
    Dim oField As Field
    Dim oStory As range
    '    On Error Resume Next
    For Each oStory In ActiveDocument.StoryRanges
        ' This goes into headers and footers as well as the regular document
        Do
            For Each oField In oStory.Fields
                If oField.Type = wdFieldRef Then
                    oField.Update
                End If
            Next oField
            Set oStory = oStory.NextStoryRange
        Loop Until oStory Is Nothing
    Next oStory
    Set oStory = Nothing
    On Error GoTo -1
End Sub

If necessary, add oField.Locked = False as a line before oField.Update. Here are Word MVP Graham Mayor's Instructions on Installing Macros found in forums.